算子标度的信息几何
Information geometry of operator scaling
摘要：
矩阵标度是一个经典的问题，有着广泛的应用。从经典信息几何的观点来看，矩阵缩放的Sinkhorn算法被解释为交替e投影。近年来，在数学和计算机科学的各个领域中，出现了一种将矩阵标度推广到完全正映射的算子标度，Sinkhorn算法也被推广到算子标度。本文从量子信息几何的观点出发，通过完全正映射的Choi表示，研究了算子Sinkhorn算法。算子Sinkhorn算法与对称对数导数度量的交替e-投影相一致，该度量是与量子估值理论相关的量子态空间上的黎曼度量.利用正定锥上不同的信息几何结构，还提出了其它类型的交替e-投影算法。
英文摘要：
Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. Other types of alternating e-projections algorithms are also provided by using different information geometric structures on the positive definite cone.
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